P1S has been used since long before LASORS was invented. LASORS is only a "guidance" document that has no formal status whatsoever. "P1S" has been an abbreviation for "PIC U/S" for many years, and as you said yourself:
P1S isnt an abbreviation it's an incorrect entry. Before LASORS P1S has only even been used incorrectly by pilots who never read the original AIC which described the required entry as P1 u/s. The mistake was made by a well known publisher of log books who published the incorrect P1s in the front of the log book.The CAA have never ever used P1s in any document. Yes you can put anything in your logbook but as an instructor you should know what entries are required for the grant or renewal of a licence or rating.